1. 1 Structures and Strategies for State Space Search 3 3.0Introduction 3.1Graph Theory 3.2Strategies for State Space Search 3.3Using the State Space to Represent Reasoning with the Predicate Calculus 3.4Epilogue and References 3.5Exercises

2. 2 Chapter Objectives Learn the basics of state space representation Learn the basics of search in state space The agent model: Has a problem, searches for a solution.

3. 3 The city of Königsberg The city is divided by a river. There are two islands at the river. The first island is connected by two bridges to both riverbanks and is also connected by a bridge to the other island. The second island two bridges each connecting to one riverbank.

4. 4 The city of Königsberg

5. 5 Graph of the Königsberg bridge system

6. 6 A labeled directed graph

7. 7 A rooted tree, exemplifying family relationships

8. 8 Definition of a graph A graph consists of A set of nodes (can be infinite) A set of arcs that connect pairs of nodes. An arc is an ordered pair, e.g., (a, b). If a directed arc connects N and M, N is called the parent, and M is called the child. If N is also connected to K, M and K are siblings. A rooted tree has a unique node which has no parents. The edges in a rooted tree are directed away from the root. Each node in a rooted tree has a unique parent.

9. 9 Definition of a graph (cont’d) A leaf or tip node is a node that has no children (sometimes also called a dead end). A path of length n is an ordered sequence of n+1 nodes such that the graph contains arcs from each node to the following ones. E.g., [a b e] is a path of length 2. On a path in a rooted graph, a node is said to be an ancestor of all the nodes positioned after it (to its right), as well as a descendant of all nodes before it (to its left).

10. 10 Definition of a graph (cont’d) A path that contains any node more than once is said to contain a cycle or loop. A tree is a graph in which there is a unique path between every pair of nodes. Two nodes are said to be connected if a path exists that includes them both.

11. 11 State space of the 8-puzzle generated by “move blank” operations

12. 12

13. 13 An instance of the traveling salesperson problem

14. 14 Search of the traveling salesperson problem. (arc label = cost from root)

15. 15 Nearest neighbor path

16. 16 Goal-directed search

17. 17 Data-directed search

18. 18

19. 19 Trace of backtracking search (Fig. 3.12)

20. 20 A trace of backtrack on the graph of Fig. 3.12

21. 21 Graph for BFS and DFS (Fig. 3.13)

22. 22 Breadth_first search algorithm

23. 23 Trace of BFS on the graph of Fig. 3.13

24. 24 Graph of Fig. 3.13 at iteration 6 of BFS

25. 25 Depth_first_search algorithm

26. 26 Trace of DFS on the graph of Fig. 3.13

27. 27 Graph of Fig. 3.13 at iteration 6 of DFS

28. 28 BFS, label = order state was removed from OPEN

29. 29 DFS with a depth bound of 5, label = order state was removed from OPEN